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openAI-math-dataset / lean /OAI /Probability /GaussianInformation /Kernel /ProjectionMoments.lean
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import Mathlib
import OAI.Probability.GaussianInformation.Kernel.GaussianImages
namespace OAI
open MeasureTheory ProbabilityTheory Metric Set
open scoped ENNReal
namespace KernelPotential
open Filter
open scoped Topology NNReal
namespace Projection
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal Topology
open InnerProductSpace Matrix MeasureTheory Set
open scoped RealInnerProductSpace ENNReal
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal NNReal
namespace Support
open Metric Matrix WithLp
noncomputable def gaussianRows (k d : ℕ) : Measure (Fin k → Fin d → ℝ) :=
Measure.pi (fun _ => Measure.pi (fun _ => gaussianReal 0 1))
noncomputable def projectVector {k d : ℕ} (X : Fin k → Fin d → ℝ)
(v : EuclideanSpace ℝ (Fin d)) : Fin k → ℝ := fun r => ∑ j, X r j * v j
lemma continuous_projectVector {k d : ℕ} :
Continuous (fun p : (Fin k → Fin d → ℝ) × EuclideanSpace ℝ (Fin d) => projectVector p.1 p.2) := by
unfold projectVector
fun_prop
lemma projectVector_sub {k d : ℕ} (X : Fin k → Fin d → ℝ)
(v w : EuclideanSpace ℝ (Fin d)) : projectVector X (v-w) = projectVector X v-projectVector X w := by
ext r
simp only [projectVector, PiLp.sub_apply, mul_sub, Finset.sum_sub_distrib, Pi.sub_apply]
lemma projectVector_rowMatrix {k q d : ℕ} (X : Fin k → Fin d → ℝ)
(v : Fin q → EuclideanSpace ℝ (Fin d)) (r : Fin k) (i : Fin q) :
projectVector X (v i) r = (rowMatrix v *ᵥ X r) i := by
simp only [projectVector, mulVec, dotProduct, rowMatrix]
apply Finset.sum_congr rfl
intro j _
ring
lemma boxScale_cancel (k q : ℕ) (η : ℝ) (hη : 0 < η) :
boxScale k η ^ q * ENNReal.ofReal ((4*η)^(q*k)) = ENNReal.ofReal ((2:ℝ)^(k*q)) := by
have h2 : 0 < 2*η := by positivity
have he : (4*η)^(q*k) = (2:ℝ)^(k*q)*((2*η)^k)^q := by
rw [show (4:ℝ)*η=2*(2*η) by ring, mul_pow, ← pow_mul, Nat.mul_comm q k]
rw [he, ENNReal.ofReal_mul (pow_nonneg (by positivity) _),
ENNReal.ofReal_pow (pow_nonneg h2.le _), boxScale, ← ENNReal.inv_pow,
mul_left_comm, ENNReal.inv_mul_cancel (by positivity) (by finiteness), mul_one]
lemma fixedReplica_moment_le {k q d : ℕ} (hk : 0 < k)
(η : ℝ) (hη : 0 < η) (v : Fin (q+1) → EuclideanSpace ℝ (Fin d)) :
(∫⁻ p : (Fin k → Fin d → ℝ) × (Fin k → ℝ),
∏ i, boxKernel η (projectVector p.1 (v i)) p.2 ∂(gaussianRows k d).prod volume) ≤
ENNReal.ofReal ((2:ℝ)^(k*q)) * gramInverseProduct k (fun i : Fin q => v i.succ-v 0) := by
classical
let diff := fun i : Fin q => v i.succ-v 0
let D := {X : Fin k → Fin d → ℝ | ∀ (i : Fin q) r,
|projectVector X (v i.succ) r-projectVector X (v 0) r| ≤ 2*η}
have hD : MeasurableSet D := by
dsimp only [D]
simp only [Set.ofPred_forall]
apply MeasurableSet.iInter
intro i
apply MeasurableSet.iInter
intro r
exact measurableSet_le
((((measurable_pi_apply r).comp (continuous_projectVector.measurable.comp
(measurable_id.prodMk measurable_const))).sub
((measurable_pi_apply r).comp (continuous_projectVector.measurable.comp
(measurable_id.prodMk measurable_const)))).abs) measurable_const
have hdiff : D = {X : Fin k → Fin d → ℝ | ∀ r i, |(rowMatrix diff *ᵥ X r) i| ≤ 2*η} := by
ext X
simp only [D, Set.mem_ofPred_eq]
rw [forall_comm]
congr! 2
simp only [← projectVector_rowMatrix, diff, projectVector_sub, Pi.sub_apply]
have hprob : gaussianRows k d D ≤ gramInverseProduct k diff *
ENNReal.ofReal ((4*η)^(q*k)) := by
rw [hdiff, gramInverseProduct_eq_inverseVolume]
simpa only [gaussianRows, Fintype.card_fin, rowMatrix_gram, show 2*(2*η)=4*η by ring] using
gaussian_multirow_box_le_extended (ρ := Fin k) (by simpa using hk)
(rowMatrix diff) (2*η) (by positivity)
have hmeas : Measurable (fun p : (Fin k → Fin d → ℝ) × (Fin k → ℝ) =>
∏ i, boxKernel η (projectVector p.1 (v i)) p.2) := by
apply Finset.measurable_prod
intro i _
exact (measurable_boxKernel η).comp
((continuous_projectVector.measurable.comp (measurable_fst.prodMk measurable_const)).prodMk measurable_snd)
have : IsProbabilityMeasure (gaussianRows k d) := by unfold gaussianRows; infer_instance
rw [lintegral_prod _ hmeas.aemeasurable]
calc
_ ≤ ∫⁻ X, D.indicator (fun _ => boxScale k η ^ q) X ∂gaussianRows k d := by
apply lintegral_mono
intro X
simpa only [Set.indicator, D, Set.mem_ofPred_eq] using
lintegral_boxKernel_product_le η hη (fun i => projectVector X (v i))
_ = boxScale k η ^ q * gaussianRows k d D := by
rw [lintegral_indicator hD, lintegral_const, Measure.restrict_apply_univ]
_ ≤ boxScale k η ^ q * (gramInverseProduct k diff * ENNReal.ofReal ((4*η)^(q*k))) :=
mul_le_mul_right hprob _
_ = ENNReal.ofReal ((2:ℝ)^(k*q)) * gramInverseProduct k diff := by
rw [mul_left_comm, boxScale_cancel k q η hη, mul_comm]
noncomputable def smoothedProjection {α : Type*} [MeasurableSpace α] {k d : ℕ}
(μ : Measure α) (a : α → EuclideanSpace ℝ (Fin d)) (η : ℝ)
(p : (Fin k → Fin d → ℝ) × (Fin k → ℝ)) : ℝ≥0∞ :=
∫⁻ z, boxKernel η (projectVector p.1 (a z)) p.2 ∂μ
lemma measurable_projection_kernel {α : Type*} [MeasurableSpace α] {k d : ℕ}
(a : α → EuclideanSpace ℝ (Fin d)) (ha : Measurable a) (η : ℝ) :
Measurable (fun p : ((Fin k → Fin d → ℝ) × (Fin k → ℝ)) × α =>
boxKernel η (projectVector p.1.1 (a p.2)) p.1.2) :=
(measurable_boxKernel η).comp
((continuous_projectVector.measurable.comp
((measurable_fst.comp measurable_fst).prodMk (ha.comp measurable_snd))).prodMk
(measurable_snd.comp measurable_fst))
lemma measurable_smoothedProjection {α : Type*} [MeasurableSpace α] {k d : ℕ}
(μ : Measure α) [SFinite μ] (a : α → EuclideanSpace ℝ (Fin d))
(ha : Measurable a) (η : ℝ) : Measurable (smoothedProjection (k := k) μ a η) :=
(measurable_projection_kernel (k := k) (d := d) a ha η).lintegral_prod_right
lemma measurable_replicated {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] {m : ℕ}
(f : β × α → ℝ≥0∞) (hf : Measurable f) :
Measurable (fun x : β × (Fin m → α) => ∏ i, f (x.1,x.2 i)) := by
apply Finset.measurable_prod
intro i _
exact hf.comp (measurable_fst.prodMk ((measurable_pi_apply i).comp measurable_snd))
lemma measurable_replica_kernel {α : Type*} [MeasurableSpace α] {k d m : ℕ}
(a : α → EuclideanSpace ℝ (Fin d)) (ha : Measurable a) (η : ℝ) :
Measurable (fun x : ((Fin k → Fin d → ℝ) × (Fin k → ℝ)) × (Fin m → α) =>
∏ i, boxKernel η (projectVector x.1.1 (a (x.2 i))) x.1.2) :=
measurable_replicated _ (measurable_projection_kernel (k := k) (d := d) a ha η)
lemma smoothedProjection_moment_le {α : Type*} [MeasurableSpace α] {k d : ℕ}
(μ : Measure α) [SigmaFinite μ] (hμ : μ univ ≤ 1)
(a : α → EuclideanSpace ℝ (Fin d)) (ha : Measurable a) (hk : 0 < k)
(p : ℕ) (M : ℝ≥0∞)
(hcond : ∀ V : AffineSubspace ℝ (EuclideanSpace ℝ (Fin d)),
(V : Set (EuclideanSpace ℝ (Fin d))).Nonempty →
Module.finrank ℝ V.direction ≤ p →
(∫⁻ z, (ENNReal.ofReal (infDist (a z) (V : Set (EuclideanSpace ℝ (Fin d)))))⁻¹ ^ k ∂μ) ≤ M)
(q : ℕ) (hq : q ≤ p+1) (η : ℝ) (hη : 0 < η) :
(∫⁻ x, smoothedProjection (k := k) μ a η x ^ (q+1)
∂(gaussianRows k d).prod volume) ≤ ENNReal.ofReal ((2:ℝ)^(k*q))*M^q := by
have : IsProbabilityMeasure (gaussianRows k d) := by unfold gaussianRows; infer_instance
have hm := measurable_replica_kernel (k := k) (d := d) (m := q+1) a ha η
calc
_ = ∫⁻ x, ∫⁻ z : Fin (q+1) → α,
∏ i, boxKernel η (projectVector x.1 (a (z i))) x.2
∂Measure.pi (fun _ => μ) ∂(gaussianRows k d).prod volume := by
apply lintegral_congr
intro x
exact (lintegral_pi_product μ _ ((measurable_projection_kernel (k := k) (d := d) a ha η).comp
(measurable_const.prodMk measurable_id)) (q+1)).symm
_ = ∫⁻ z : Fin (q+1) → α, ∫⁻ x,
∏ i, boxKernel η (projectVector x.1 (a (z i))) x.2
∂(gaussianRows k d).prod volume ∂Measure.pi (fun _ => μ) :=
lintegral_lintegral_swap hm.aemeasurable
_ ≤ ∫⁻ z : Fin (q+1) → α, ENNReal.ofReal ((2:ℝ)^(k*q)) *
gramInverseProduct k (fun i : Fin q => a (z i.succ)-a (z 0))
∂Measure.pi (fun _ => μ) := lintegral_mono fun z =>
fixedReplica_moment_le hk η hη (fun i => a (z i))
_ = ENNReal.ofReal ((2:ℝ)^(k*q)) * ∫⁻ z : Fin (q+1) → α,
gramInverseProduct k (fun i : Fin q => a (z i.succ)-a (z 0))
∂Measure.pi (fun _ => μ) := by
apply lintegral_const_mul
exact (measurable_gramInverseProduct k).comp
(Measurable.of_eval fun i => (ha.comp (measurable_pi_apply i.succ)).sub
(ha.comp (measurable_pi_apply 0)))
_ ≤ ENNReal.ofReal ((2:ℝ)^(k*q))*M^q := mul_le_mul_right
(lintegral_replicaInverse_le μ hμ a ha k p M hcond q hq) _
end Support
open Support _root_.OAI.KernelPotential.Geometry
lemma measurable_zero_kernel {k d : ℕ} (η : ℝ) :
Measurable (fun x : (Fin k → Fin d → ℝ) × EuclideanSpace ℝ (Fin d) =>
boxKernel η (projectVector x.1 x.2) 0) :=
(measurable_boxKernel η).comp
(continuous_projectVector.measurable.prodMk measurable_const)
end Projection
end KernelPotential
end OAI